This proposal intends to develop novel mathematical methods for (i) inferring hidden'' low-dimensional or low-rank structure in neural data, (ii) testng long-standing ideas about how networks are organized in some brain areas. The theory will be both tested and guided by the analysis of electrophysiological recordings of hippocampal and cortical neurons. To do so, the PI will build on his established record of successful collaborations with experimental neuroscientists. The brain is a vast collection of interconnected neural circuits whose computations are often accomplished by local networks. Understanding the structure of these local circuits, however, is still very much a work in progress. Experimental technology in neuroscience is making rapid progress towards amassing large databases of the activity patterns and connection patterns between neurons. How to interpret this data, and what it can tell us about the structure of neural circuits, is still very much unclear. The biggest limitation is perhaps the lack of theoretical frameworks for characterizing meaningful structure in local neural circuits. Biological data often comes in the form of a matrix W (such as a matrix of pairwise correlations or synaptic weights) whose entries are likely to be measured in unnatural units that obscure the underlying structure. In other words, each entry of matrix W is a result of a non-linear transformation W=f(D)+noise of a different matrix D, where f is an unknown (but monotone) function, and the hidden structure in D may be of primary interest. It turns out that i the function f is monotone, key structural properties of D can be reliably inferred from W using tools from topological data analysis. This is because these tools measure structure that is invariant under monotone transformations. The central goals of this proposal are (1) to develop tools for detection of hidden structure in neural and other biological data using tools from computational topology and commutative algebra, and (2) to use these tools to test long-standing hypotheses about the organization of local networks in the brain.